Robust Static $\mathcal{H}_{\infty}$ Output-Feedback Control Using Polynomial Chaos

Wan, Yiming; Shen, Dongying Erin; Lucia, Sergio; Findeisen, Rolf; Braatz, Richard D.

doi:10.23919/acc.2018.8431074

Cited by 5 publications

(19 citation statements)

References 10 publications

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“…This property does not hold for the PCE-transformed approximation given in [6], [19], [23]. Moreover, it is proved by (49)-(52) in Appendix C that the worst-case upper bound for the approximation error x(t, ξ)− Φ x (ξ)X a (t) from (23) is smaller than the one obtained from the PCE-transformed system in [6], [19], [23].…”

Section: Pce-transformed Closed-loop Systemmentioning

confidence: 95%

“…Note that X a (t) and Z a (t) in ( 23) are approximates of X(t) in ( 18) and Z nom (t) in (22a). This PCE-transformed system ( 23) is different from the existing ones in two aspects: (i) The existing approach in [6], [19], [23] separately transforms the four system equations in ( 1) and ( 2) by applying standard Galerkin projection, to construct a PCE-transformed closed-loop system. In this case, if the open-loop system is unstable due to its parametric uncertainties, the divergent open-loop system state as a function of ξ would not admit a PCE.…”

Section: Pce-transformed Closed-loop Systemmentioning

confidence: 99%

“…In contrast, our proposed transformation directly considers the original closed-loop dynamics, which allows PCEs of system states by stabilizing the closed-loop dynamics. (ii) With the error term Rx (t) removed from (18a), the squared L 2 -induced gain from w(t) to z nom (t, ξ) in (18a) and (16d) is equal to the squared H ∞ -norm of the PCE-transformed approximation (23), i.e., sup…”

Section: Pce-transformed Closed-loop Systemmentioning

confidence: 99%

“…In this paper, a PCE-based H ∞ static output-feedback (SOF) control approach is proposed for uncertain linear time-invariant (LTI) systems with polynomial dependence on probabilistic time-invariant parametric uncertainties. The proposed PCE-based transformation explicitly copes with the closed-loop dynamics, and preserves the L 2 -induced gain, which results in a smaller approximation error compared to existing PCE-transformed systems reported in [19], [23].…”

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

A Polynomial Chaos Approach to Robust Static Output-Feedback Control With Bounded Truncation Error

Wan

Shen

Lucia

et al. 2023

IEEE Trans. Automat. Contr.

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This article considers the H∞ static outputfeedback control for linear time-invariant uncertain systems with polynomial dependence on probabilistic time-invariant parametric uncertainties. By applying polynomial chaos theory, the control synthesis problem is solved using a highdimensional expanded system which characterizes stochastic state uncertainty propagation. A closed-loop polynomial chaos transformation is proposed to derive the closed-loop expanded system. The approach explicitly accounts for the closed-loop dynamics and preserves the L2-induced gain, which results in smaller transformation errors compared to existing polynomial chaos transformations. The effect of using finite-degree polynomial chaos expansions is first captured by a normbounded linear differential inclusion, and then addressed by formulating a robust polynomial chaos based control synthesis problem. This proposed approach avoids the use of high-degree polynomial chaos expansions to alleviate the destabilizing effect of truncation errors, which significantly reduces computational complexity. In addition, some analysis is given for the condition under which the robustly stabilized expanded system implies the robust stability of the original system. A numerical example illustrates the effectiveness of the proposed approach.

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Section: Pce-transformed Closed-loop Systemmentioning

confidence: 95%

Section: Pce-transformed Closed-loop Systemmentioning

confidence: 99%

Section: Pce-transformed Closed-loop Systemmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

A Polynomial Chaos Approach to Robust Static Output-Feedback Control With Bounded Truncation Error

Wan

Shen

Lucia

et al. 2023

IEEE Trans. Automat. Contr.

Self Cite

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show abstract

“…Recent studies have already suggested using the GPC formalism to solve robust control problems in a probabilistic framework, that is, to determine a deterministic controller guaranteeing or optimizing the expected value of a particular closed‐loop performance criterion computed over the probability distribution of the plant uncertainty. These include References 6,7 for LQ control, References 8,9 for H 2 and H ∞ control, and References 10,11 for receding horizon control.…”

Section: Introductionmentioning

confidence: 99%

Stochastic linear quadratic control via random parameter‐dependent truncated balanced realization

Nechak

Raynaud

Kulcsár

2020

Intl J Robust & Nonlinear

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This article deals with stochastic linear quadratic (LQ) control by using random parameter‐dependent truncated balanced realization (RPD‐TBR). The main contribution of the article is theoretical formulation to approximate high‐order stochastic LQ control problems, using the RPD‐TBR instead of the original RPD full state model. The whole approach uses the nonintrusive and intrusive generalized polynomial chaos (GPC) approximations to generate an RPD‐TBR and the corresponding reduced‐order stochastic LQ control formulation. The latter is then rewritten in the GPC space using an intrusive Galerkin‐type projection. Solving the reduced‐order Riccati equation for the resulting deterministic LQ problem enables to suitably approximate the solution of the original full‐order stochastic LQ control problem. The feasibility and effectiveness of the proposed approach are assessed by considering the LQ control of the tip‐tilt mirror for an Extremely Large Telescope‐class adaptive optics system, when the actuator's dynamics depends on an uncertain parameter with known distribution. Through numerical simulations, the proposed approach ability to perform LQ gain selection and performance evaluation for plants with probabilistic uncertainty more efficiently than the standard Monte Carlo approach is illustrated.

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Robust controller design of a semi-active quasi-zero stiffness air suspension based on polynomial chaos expansion

Jiang

Liang

et al. 2023

Journal of Vibration and Control

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Air suspension is one of the heart of electrified chassis, which plays a key role in vehicle ride comfort, driving stability and safety. In order to further improve the suspension performance to meet the demand of complex and changing road conditions, various new suspension structures have emerged. This paper inherits the advantages of electronically controlled air suspension and proposes a quasi-zero stiffness air suspension system combined with pneumatic negative stiffness mechanism. Aiming at the random uncertainty of the structural parameters of the suspension system due to manufacturing, use environment and wear, a robust controller for a semi-active quasi-zero stiffness air suspension based on polynomial chaos expansion (PCE) is designed. The quasi-zero stiffness air suspension alternative model is established by the coefficient state equation of PCE, and the state feedback control law considering the random parameter uncertainty is obtained based on the H2 control theory solution. Simulations and HiL tests are conducted to verify the effectiveness and real-time performance of proposed PCE-H2 controller. Simulation results show that the performance of PCE-H2 control law is better than that of the conventional H2. The HiL results also show that the PCE-H2 control law considering uncertainty is 3.2% less effective than the conventional H2 control law in overall performance of the suspension. Besides, the control effect of the PCE-H2 control law is 38.7% better in the performance maintenance of the semi-active suspension.

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Robust Static $\mathcal{H}_{\infty}$ Output-Feedback Control Using Polynomial Chaos

Cited by 5 publications

References 10 publications

A Polynomial Chaos Approach to Robust Static Output-Feedback Control With Bounded Truncation Error

A Polynomial Chaos Approach to Robust Static Output-Feedback Control With Bounded Truncation Error

Stochastic linear quadratic control via random parameter‐dependent truncated balanced realization

Robust controller design of a semi-active quasi-zero stiffness air suspension based on polynomial chaos expansion

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